Analytic Besov Spaces Research Articles

Operators Induced by Radial Measures Acting on the Dirichlet Space

Let {mathbb {D}} be the unit disc in the complex plane. Given a positive finite Borel measure mu on the radius [0, 1), we let mu _n denote the n-th moment of mu and we deal with the action on spaces of analytic functions in {mathbb {D}} of the operator of Hibert-type {mathcal {H}}_mu and the operator of Cesàro-type {mathcal {C}}_mu which are defined as follows: If f is holomorphic in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), then {mathcal {H}}_mu (f) is formally defined by {mathcal {H}}_mu (f)(z) = sum _{n=0}^infty left( sum _{k=0}^infty mu _{n+k}a_kright) z^n (zin {mathbb {D}}) and {mathcal {C}}_mu (f) is defined by mathcal C_mu (f)(z) = sum _{n=0}^infty mu _nleft( sum _{k=0}^na_kright) z^n (zin {mathbb {D}}). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in {mathbb {D}}. In this paper we study the action of the operators {mathcal {H}}_mu and {mathcal {C}}_mu on the Dirichlet space {mathcal {D}} and, more generally, on the analytic Besov spaces B^p (1le p<infty ).

On Stevic-Sharma operators from the analytic Besov space into Bloch-type spaces

On Stevic-Sharma operators from the analytic Besov space into Bloch-type spaces

Sums of generalized weighted composition operators from the analytic Besov space into the Bloch space

Sums of generalized weighted composition operators from the analytic Besov space into the Bloch space

Analytic Besov spaces, approximation, and closed ideals

AbstractIn this paper, we give a complete description of closed ideals of the Banach algebra $\mathcal {B}^{s}_{p}\cap \lambda _{\alpha }$ , where $\mathcal {B}^{s}_{p}$ denotes the analytic Besov space and $\lambda _{\alpha }$ is the separable analytic Lipschitz space. Our result extends several previous results in Bahajji-El Idrissi and El-Fallah (2020, Studia Mathematica 255, 209–217), Bouya (2009, Canadian Journal of Mathematics 61, 282–298), and Shirokov (1982, Izv. Ross. Akad. Nauk Ser. Mat. 46, 1316–1332).

Approximation in Weighted Analytic Besov Spaces and in Generalized Fock Spaces

Approximation in Weighted Analytic Besov Spaces and in Generalized Fock Spaces

Essential Norm of Difference of Composition Operators from Analytic Besov Spaces to Bloch Type Spaces

Essential Norm of Difference of Composition Operators from Analytic Besov Spaces to Bloch Type Spaces

Differences of composition operators from analytic Besov spaces into little Bloch type spaces

We present some characterizations for the compactness of the difference of two composition operators acting between analytic Besov spaces and the weighted little Bloch type space over the unit disk.

A Radial Integrability Result Concerning Bounded Functions in Analytic Besov Spaces with Applications

We prove that for every $p\ge 1$ there exists a bounded function in the analytic Besov space $B^p$ whose derivative is "badly integrable", along every radius. We apply this result to study multipliers and weighted superposition operators acting on the spaces $B^p$.

Composition Operators on Some Banach Spaces of Harmonic Mappings

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.

Toeplitz and Hankel operators from Bergman to analytic Besov spaces of tube domains over symmetric cones

We characterize bounded Toeplitz and Hankel operators from weighted Bergman spaces to weighted Besov spaces in tube domains over symmetric cones. We deduce weak factorization results for some Bergman spaces of this setting.

Essential Norm of Weighted Composition Operators from Analytic Besov Spaces into Zygmund Type Spaces

In this paper, we give some estimates for the essential norm of weighted composition operators from analytic Besov spaces into Zygmund type spaces. In particular, a new characterization for the boundedness and compactness of the weighted composition operators uCϕ is obtained.

Toeplitz operators with pluriharmonic symbol on the unit ball

A well-known result going back to M. Engliš shows that Toeplitz operators with bounded symbol on the Bergman space A2(D) cannot be characterized by a Brown–Halmos type operator identity. In the opposite direction we prove that Toeplitz operators with pluriharmonic symbol on the analytic functional Hilbert spaces Hm(B) with reproducing kernel Km(z,w)=(1−〈z,w〉)−m can be characterized by an operator identity that specializes to the Brown–Halmos condition in the case of the Hardy space on the unit disc. We thus extend results obtained by Louhichi and Olofsson for the standard weighted Bergman spaces on the unit disc to the analytic Besov spaces Hm(B) on the unit ball.

A generalized Hilbert operator acting on conformally invariant spaces

If μ is a positive Borel measure on the interval [0,1), we let Hμ be the Hankel matrix Hμ=(μn,k)n,k≥0 with entries μn,k=μn+k, where, for n=0,1,2,…, μn denotes the moment of order n of μ. This matrix formally induces the operator Hμ(f)(z)=∑n=0∞(∑k=0∞μn,kak)zn on the space of all analytic functions f(z)=∑k=0∞akzk, in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators Hμ on Hardy spaces has been recently studied. This article is devoted to a study of the operators Hμ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the Qs-spaces.

Weighted Composition Operators from Analytic Besov Spaces into the Bloch Space

In this paper, we give some new essential norm estimates of weighted composition operators \(uC_{\varphi }\) from analytic Besov spaces into the Bloch space, where u is a function analytic on the unit disk \(\mathbb {D}\) and \(\varphi \) is an analytic self-map of \(\mathbb {D}\). Moreover, new characterizations for the boundedness, compactness and essential norm of weighted composition operators \(uC_{\varphi }\) are obtained by the nth power of the symbol \(\varphi \) and the Volterra operators \(I_u\) and \(J_u\).

Norm, essential norm and weak compactness of weighted composition operators between dual Banach spaces of analytic functions

In this paper we estimate the norm and the essential norm of weighted composition operators from a large class of – non-necessarily reflexive – Banach spaces of analytic functions on the open unit disk into weighted type Banach spaces of analytic functions and Bloch type spaces. We also show the equivalence of compactness and weak compactness of weighted composition operators from these weighted type spaces into a class of Banach spaces of analytic functions, that includes a large family of conformally invariant spaces like BMOA and analytic Besov spaces.

AN INTEGRAL REPRESENTATION FOR BESOV AND LIPSCHITZ SPACES

AbstractIt is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation $$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$ where ${\it\mu}$ is a complex Borel measure with $|{\it\mu}|(\mathbb{D})&lt;\infty$. We generalize this result to all Besov spaces $B_{p}$ with $0&lt;p\leq 1$ and all Lipschitz spaces ${\rm\Lambda}_{t}$ with $t&gt;1$. We also obtain a version for Bergman and Fock spaces.

WEIGHTED COMPOSITION OPERATORS ON THE MINIMAL MÖBIUS INVARIANT SPACE

Abstract. We will characterize the boundedness and compactness ofweighted composition operators on the minimal Mobius invariant space. 1. IntroductionHere and henceforth, Dwill denote the open unit disk D:= {z ∈ C: |z| < 1}.The set of all conformal automorphisms of Dforms a group, called a Mo¨biusgroup and denoted by Aut(D). For any λ ∈ D, letα λ (z) =λ −z1−λzbe the Mo¨biustransformationof D. LetX be alinearspaceofanalyticfunctionson D. Then X is said to be Mo¨bius invariant if f ◦α ∈ X for all f ∈ X and allα ∈Aut(D). A typical example of Mo¨bius invariant spaces is the Besov space.For 1 < p < ∞, let B p be the space of analytic functions f on Dsuch thatZ D |f ′ (z)| p (1−|z|) p−2 dA(z) < ∞,where dA is the normalized Lebesgue area measure on D. Then B p is theBanach space with the normkfk B p = |f(0)|+Z D |f ′ (z)| p (1 −|z|) p−2 dA(z) 1/p .If p = 2, B 2 is the classical Dirichlet space that is minimal as Mo¨bius invariantHilbert space of analytic functions on D. The analytic Besov space B

Characterization of weighted analytic Besov spaces in terms of operators of fractional differentiation

Abstract Let $$\mathbb{D}$$ stand for the unit disc in the complex plane ℂ. Given 0 &lt; p &lt; ∞, −1 &lt; λ &lt; ∞, the analytic weighted Besov space $$B_p^\lambda \left( \mathbb{D} \right)$$ is defined to consist of analytic in $$\mathbb{D}$$ functions such that $$\int\limits_\mathbb{D} {\left( {1 - \left| z \right|^2 } \right)^{Np - 2} \left| {f^{\left( N \right)} \left( z \right)} \right|^p d\mu _\lambda \left( z \right) &lt; \infty ,}$$ where dμ λ(z) = (λ + 1)(1 − |z|2)λ dμ(z), $$d\mu (z) = \tfrac{1} {\pi }dxdy$$, and N is an arbitrary fixed natural number, satisfying N p &gt; 1 − λ. We provide a characterization of weighted analytic Besov spaces $$B_p^\lambda \left( \mathbb{D} \right)$$, 0 &lt; p &lt; ∞, in terms of certain operators of fractional differentiation R zα,t of order t. These operators are defined in terms of construction known as Hadamard product composition with the function b. The function b is calculated from the condition that R zα,t (uniquely) maps the weighted Bergman kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha }$$ to the similar (weight parameter shifted) kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha - t}$$, t &gt; 0. We also show that $$B_p^\lambda \left( \mathbb{D} \right)$$ can be thought as the image of certain weighted Lebesgue space $$L^p \left( {\mathbb{D},d\nu _\lambda } \right)$$ under the action of the weighted Bergman projection $$P_\mathbb{D}^\alpha$$.

Compact and weakly compact composition operators from the Bloch space into Möbius invariant spaces

We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like BMOA, Qα, and analytic Besov spaces Bp. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding “small” spaces we also characterize the boundedness and show that it is equivalent to compactness.

Isometric composition operators on the analytic Besov spaces

We investigate the isometric composition operators on the analytic Besov spaces. For 1<p<2 we show that an isometric composition operator is induced only by a rotation of the disk. For p>2, we extend previous work on the subject. Finally, we analyze this same problem for the Besov spaces with an equivalent norm.